Principal component pursuit via pcpr

Pattern recognition in environmental mixtures analyses

In this document, we illustrate how to employ Principal Component Pursuit (PCP) for pattern recognition in environmental health (EH) analyses with the pcpr R package. Additional information regarding pcpr’s statistical methodology and computational details are provided in Chillrud et al. (2022).

1 Overview

1.1 The mixtures model

PCP models an observed exposure matrix \(D\) as the sum of three underlying ground-truth matrices: \[ \underset{\text{mixture}}{D_{n \times p}} = \underset{\text{low-rank}}{L_0} + \underset{\text{sparse}}{S_0} + \underset{\text{noise}}{Z_0} \] a low-rank matrix \(L_0\) encoding consistent patterns of exposure, a sparse matrix \(S_0\) isolating unique or outlying exposure events (that cannot be explained by the consistent exposure patterns), and dense noise \(Z_0\). All of these matrices are of dimension \(n \times p\), where \(n\) is the number of observations (e.g., study participants or measurement dates) and \(p\) is the number of exposures (chemical and/or non-chemical stressors). Beyond this mixtures model, the main (EH-relevant) assumption made by PCP is that \(Z_0 \sim \mathcal{N}(\mu, \sigma^2)\) consists of i.i.d. Gaussian noise corrupting each entry of the overall exposure matrix \(D\). A visual example of PCP’s mixtures model is provided below:

A visual example of the PCP mixtures model.
A visual example of the PCP mixtures model.

Under these minimal assumptions, PCP aims to decompose the observed mixture \(D\) into the above mixtures model via an optimization program, thereby recovering the ground-truth \(L_0\) and \(S_0\) with accurate estimates \(\hat{L}\) and \(\hat{S}\).

1.2 The low-rank matrix

The estimated low-rank matrix \(\hat{L}\) provides information on the consistent exposure patterns, satisfying \[ r = \text{rank}(\hat{L}) \ll \min(n, p). \] The rank \(r\) corresponds to the (relatively few) number of underlying patterns governing the mixture, such as specific sources or behaviors leading to exposure. Notice that \(\hat{L} \in \mathbb{R}^{n \times p}\), meaning it is still defined in terms of the original variables. Put differently, \(\hat{L}\) can be taken as a robust approximation to the true environmental mixture matrix, unperturbed by outliers (captured in \(\hat{S}\)) or noise (handled by PCP’s noise term). In this way, the latent exposure patterns are encoded in \(\hat{L}\) rather than directly estimated. To explicitly obtain the exposure patterns from \(\hat{L}\), PCP may then be paired with various matrix factorization methods (e.g., PCA, factor analysis, or NMF) that yield chemical loadings and individual scores for use in downstream health models.

1.3 The sparse matrix

The estimated sparse matrix \(\hat{S}\) captures unusually high/low outlying exposure events, unexplained by the identified patterns in \(\hat{L}\). Most entries in \(\hat{S}\) are 0, with non-zero entries identifying such extreme exposure activity. The number, location (i.e., support), and value of non-zero entries in \(\hat{S}\) need not be defined; PCP isolates these during optimization.

2 PCP algorithms

root_pcp() RRMC()
Convex? Yes No
Convergence? Slow Fast
Expected low-rank structure? Well-defined Messy
Parameters? \(D, \lambda, \mu\) \(D, r, \eta\)
Supports LOD penalty? Yes Yes
Supports non-negativity constraint? Yes No
Rank determination? Autonomous User-defined
Solver? ADMM Iterative rank-based

2.1 Root PCP

The convex model available in pcpr is Root PCP, denoted here as \(\sqrt{\text{PCP}}\). For a comprehensive technical understanding, we refer readers to Zhang et al. (2021) introducing \(\sqrt{\text{PCP}}\). Here we briefly introduce the root_pcp function in pcpr.

2.2 RRMC

3 Environmental health example analysis: PM2.5 source apportionment

Here we will demonstrate how to apply PCP to environmental mixtures data using an example air pollution source apportionment analysis. Environmental health researchers often aim to identify sources that drive potentially harmful environmental exposures. Investigating potential associations between the identified sources, or patterns of exposure, and adverse health outcomes can then help pave the way toward targeted interventions or public health policy recommendations.

For our analysis, we aim to apportion speciated PM\(_{2.5}\) to its sources using the queens dataset that comes with the pcpr R package. The queens dataset consists of real chemical concentrations (in µg/m\(^3\)) of 26 species of PM\(_{2.5}\) measured every three to six days from 04/04/2001 through 12/30/2021 by an EPA AQS air monitor located in Queens, New York City.

We’ll begin by exploring the raw queens data - corresponding to the \(D\) matrix in our formal mixtures model above - before applying PCP to obtain estimates for \(\hat{L}\) and \(\hat{S}\). We can then compare the PCP-recovered low dimensional structure in \(\hat{L}\) with the observed raw patterns of \(D\).

3.1 Exploring the raw queens data

First, let’s load the pcpr package (along with a few other packages we will need for data processing) and take a look at the queens dataset:

library(dplyr)
library(ggplot2)
library(magrittr)
library(pcpr)
library(stringr)
library(tidyr)

queens 
## # A tibble: 2,443 × 27
##    Date            Al   NH4      As     Ba       Br     Cd      Ca      Cl
##    <date>       <dbl> <dbl>   <dbl>  <dbl>    <dbl>  <dbl>   <dbl>   <dbl>
##  1 2001-04-04 NA      1.62  NA      NA     NA       NA     NA      NA     
##  2 2001-04-07  0      2.66   0       0.012  0.00488  0      0.0401  0.0079
##  3 2001-04-13  0.0094 1.41   0.0016  0.024  0.00211  0.004  0.036   0     
##  4 2001-04-19  0.0104 1.22   0.001   0.006  0.00422  0      0.0543  0.003 
##  5 2001-04-25  0.0172 0.723  0.0024  0.015  0.00117  0      0.0398  0     
##  6 2001-05-01  0.0384 3.48   0.0017  0.041  0.00873  0.001  0.136   0     
##  7 2001-05-04  0.0964 6.22   0.0025  0.039  0.0111   0      0.137   0     
##  8 2001-05-07  0.004  0.233  0.001   0.016  0.00263  0      0.055   0.0054
##  9 2001-05-10  0.0547 2.04   0.001   0.055  0.00521  0      0.121   0.001 
## 10 2001-05-13  0.0215 0.229  0       0.021  0.00122  0      0.0249  0     
## # ℹ 2,433 more rows
## # ℹ 18 more variables: Cr <dbl>, Cu <dbl>, EC <dbl>, Fe <dbl>, Pb <dbl>,
## #   Mg <dbl>, Mn <dbl>, Ni <dbl>, OC <dbl>, K <dbl>, Se <dbl>, Si <dbl>,
## #   Na <dbl>, S <dbl>, Ti <dbl>, NO3 <dbl>, V <dbl>, Zn <dbl>

Let’s spend some time visualizing some of the trends of the raw dataset, to get a sense of what we’re dealing with. We can start by plotting each of the measured chemical species as timeseries. Plotted in black are the raw observed PM\(_{2.5}\) measurements (in µg/m\(^3\)) over time (04/04/2001 - 12/30/2021), and plotted in red are some (rough) lines of best fit:

queens %>%
  pivot_longer(
    colnames(queens)[-1], names_to = "chem", values_to = "concentration"
  ) %>%
  filter(!is.na(concentration)) %>% 
  ggplot(aes(x = Date, y = concentration)) +
  geom_line() +
  geom_smooth(color = "red", formula="y ~ x", method = "loess", span = 0.05) +
  facet_wrap(~chem, scales = "free_y") +
  labs(x = "Date", y = "Concentration (µg/m^3)") +
  theme_bw()

Some initial points to make note of, looking at the data:

  • The y-axis scale for each chemical’s concentration differs widely (e.g., NH4 ranges from 0 to 9 µg/m\(^3\), while Ti ranges from 0 to 0.04 µg/m\(^3\)). We will have to take this into account during preprocessing (which we do in the next section Preliminary PCA).
  • Some chemical species record negative measurements (e.g., Al, Ba, Cd). As explained in the EPA AQS data documentation, this is due to idiosyncrasies in the instruments used to collect the measurements.
  • We can see strong seasonal trends in some species (e.g., NO3), as well as a reduction in concentration over time in others (e.g., Ni).
  • Many species have prominent outliers, or extreme exposure events (e.g., Cr, Cu). Further, some species have extreme exposure events following seasonal trends, e.g., K.
  • Elemental carbon (EC) and organic carbon (OC) are both missing measurements for the first 8 years of the dataset, likely because the air monitors for those two species were not operational from 2001 - 2009. Handling such systematic missingness is out of scope for this tutorial, so we make the simple decision to omit all data from 2001 - 2009, rather than dropping EC and OC, since including both will be helpful in our source apportionment study.

Below we remove the early measurement dates from 2001 to 2009 that are missing observations for EC and OC, yielding the queens_small dataset that we will use moving forward:

start_date_df <- queens %>% select(Date, EC, OC) %>% na.omit() %>% slice_head(n = 1)
start_date <- start_date_df$Date[1]
cat("Start date:", as.character(start_date))
## Start date: 2009-04-01
queens_small <- queens %>% filter(Date >= as.Date(start_date))
queens_small
## # A tibble: 1,535 × 27
##    Date          Al   NH4      As    Ba     Br    Cd     Ca     Cl    Cr     Cu
##    <date>     <dbl> <dbl>   <dbl> <dbl>  <dbl> <dbl>  <dbl>  <dbl> <dbl>  <dbl>
##  1 2009-04-01 0.044 0.883 0       0     0.0034 0     0.0507 0.677  0     0.0026
##  2 2009-04-04 0.008 0.899 0       0     0.0027 0.009 0.012  0      0.001 0.001 
##  3 2009-04-07 0.03  0.623 0       0     0      0     0.0142 0.006  0     0.0039
##  4 2009-04-10 0.016 3.43  0       0     0.0063 0     0.0674 0.0561 0     0.0062
##  5 2009-04-13 0.025 0.774 0.001   0     0.003  0.012 0.0372 0.001  0     0.0025
##  6 2009-04-16 0.002 0.822 0       0.003 0.0051 0.004 0.0458 0      0     0.001 
##  7 2009-04-19 0     0.748 0.00105 0     0.0015 0     0.0299 0.0547 0.002 0.001 
##  8 2009-04-22 0     2.1   0       0     0.0033 0     0.0488 0.0101 0.002 0.0051
##  9 2009-04-25 0.044 2.47  0       0     0.0037 0     0.0626 0.024  0.001 0.0064
## 10 2009-04-28 0.045 1.33  0       0     0.0043 0     0.0494 0.004  0     0.0023
## # ℹ 1,525 more rows
## # ℹ 16 more variables: EC <dbl>, Fe <dbl>, Pb <dbl>, Mg <dbl>, Mn <dbl>,
## #   Ni <dbl>, OC <dbl>, K <dbl>, Se <dbl>, Si <dbl>, Na <dbl>, S <dbl>,
## #   Ti <dbl>, NO3 <dbl>, V <dbl>, Zn <dbl>

Next, let’s take a look at the correlation structure of our queens_small data:

queens_small %>% 
  select(-Date) %>%
  as_tibble() %>% 
  GGally::ggcorr(., method = "pairwise.complete.obs", limits = F, label = F, size = 5)

The high dimensionality of our queens_small air pollution mixture matrix gives rise to this relatively complex correlation matrix. No strong patterns jump out right away here, although Na’s correlation with Cl and Mg calls to mind sea salt from the Atlantic Ocean or perhaps road salt kicked up by traffic. We’d like to employ PCP in order to reduce the complexity of our data for more robust downstream analysis. Keep this correlation matrix in mind, since after applying PCP, we’ll examine the correlations matrix of \(\hat{L}\) and compare.

3.2 Preliminary PCA

Before applying PCP to our mixture, it’s good to first take a look at what Principal Component Analysis (PCA) is able to extract from the queens_small data. We do this for a couple of reasons:

  1. First, we’d like to establish a baseline for comparison. To better understand the effect of applying PCP to the queens_small data, we’d like to see what low dimensional structures exist in the raw data from the outset.
  2. Second, understanding the behavior of the singular values governing the raw data will help inform choices we must make pertaining to PCP’s optimization scheme. A preliminary PCA will enable us to scrutinize our data’s singular values. \(\sqrt{\text{PCP}}\) is well-suited for data exhibiting rapidly decaying singular values (e.g., imaging data), while RRMC is best for more messy data with slowly decaying singular values (environmental mixtures data typically falls into this bucket).

Before we are able to call upon PCA (or PCP, for that matter) we need to first preprocess our data for better numerical stability. In practical terms, both PCA and PCP’s statistical routines are sensitive to data with variable scales. Before arbitrarily normalizing or standardizing our data, let’s examine the distribution of our queens_small dataset:

queens_small %>%
  pivot_longer(
    colnames(queens_small)[-1], names_to = "chem", values_to = "concentration"
  ) %>%
  filter(!is.na(concentration)) %>% 
  ggplot(aes(x = concentration)) +
  geom_histogram(bins = 50) +
  theme_bw() +
  facet_wrap(~chem, scales = "free")

Most of the chemical species look as though they are normally distributed. We have many choices for how we’d like to preprocess our data, e.g., standardize the data, min-max normalization, etc. Because most of our data appears (roughly) normally distributed, and because in PCA analyses, standardization is typical, let’s go ahead and standardize (scale and center) our data to have \(\mu = 0\), \(\sigma = 1\):

queens_standardized <- queens_small %>%
  select(-Date) %>%
  scale() %>%
  as_tibble() 

queens_standardized$Date <- queens_small$Date

queens_standardized %>% 
  pivot_longer(
    colnames(queens_small)[-1], names_to = "chem", values_to = "concentration"
  ) %>%
  filter(!is.na(concentration)) %>%
  ggplot(aes(x = concentration)) +
  geom_histogram(bins = 50) +
  theme_bw() +
  facet_wrap(~chem, scales = "free")

Great, now we’re ready to run the preliminary PCA analysis! We’ll use the stats::prcomp function for this, which by default omits missing values in our dataset. Instead, let’s handle NAs by imputing with the column-wise means. Below, we define two helper functions, mean_impute and pca:

mean_impute <- function(x) {
  x[is.na(x)] <- mean(x, na.rm = T)
  x
}

pca <- function(mat, pcs = c("PC1", "PC2", "PC3", "PC4", "PC5", "PC6"), colgroups = NULL, impute = T) {
  
  if ("Date" %in% colnames(mat)) mat <- mat %>% select(-Date)
  if (impute) mat <- mat %>% apply(., 2, mean_impute)
  
  #### PCA ####
  pca.ln <- prcomp(mat) 
  
  #### VARIANCE TABLE ####
  singvalues_ln <- matrix(pca.ln$sdev^2)
  perc_variance <- round(100*matrix(pca.ln$sdev^2/sum(pca.ln$sdev^2)),1)
  
  pca_summary <- data.frame("Principle component" = 1:min(dim(mat)), "Singular values" = singvalues_ln, "Percent variance" = perc_variance, "Total cumulative variance" = purrr::accumulate(perc_variance, sum))
  
  var_tbl <- kableExtra::kbl(pca_summary, col.names = c("Principle component", "Singular value", "% variance", "Total cumulative variance"), align = "c") %>% 
    kableExtra::kable_classic(full_width = F, html_font = "Cambria", position = "center") %>% 
    kableExtra::kable_styling(bootstrap_options = c("hover", "condensed"), fixed_thead = T)
  
  #### LOADINGS ####
  pca.ln.ld <- as.data.frame.matrix(pca.ln$rotation)
  pca.ln.ld$chem <- row.names(pca.ln.ld)
  
  if (!is.null(colgroups)) {
    colgroups <- colgroups %>% dplyr::rename(chem = !!names(colgroups)[1])
  } else {
    colgroups <- data.frame(chem = colnames(mat), group = "1")
  }
  grouping <- names(colgroups)[2]
  
  plot_loadings_pca <- pca.ln.ld %>% 
    tidyr::gather(key = "PC", value = "Loading", -chem) %>% 
    tibble::as_tibble() %>% 
    dplyr::right_join(., colgroups, by = "chem")
  plot_loadings_pca$chem <- factor(as.character(plot_loadings_pca$chem), levels = unique(as.character(plot_loadings_pca$chem)))
  
  loadings <- plot_loadings_pca %>%
    dplyr::filter(PC %in% pcs) %>% 
    ggplot(aes(x = chem, y = Loading, color = !!sym(grouping))) + 
    geom_point() +
    geom_segment(aes(yend=0, xend = chem)) +
    facet_wrap(~ PC) + theme_bw() +
    theme(legend.position = "bottom", 
          axis.text.x = element_text(angle = 45, hjust = 1),
          strip.background = element_rect(fill = "white"),
          axis.title.x = element_blank(),
          axis.title.y = element_blank()) +
    geom_hline(yintercept = 0, linewidth = 0.2) + 
    ggtitle("Principle Component Loadings")
  
  list(var = var_tbl, var_df = pca_summary, load = loadings)
}

prelim_pca <- pca(queens_standardized)

Now let’s take a look at our preliminary PCA. We’ll start by examining the singular values and the proportion of variance explained by each component:

prelim_pca$var
Principle component Singular value % variance Total cumulative variance
1 5.6835249 22.3 22.3
2 2.3916708 9.4 31.7
3 2.1466041 8.4 40.1
4 1.6284138 6.4 46.5
5 1.4501119 5.7 52.2
6 1.2056250 4.7 56.9
7 1.1309203 4.4 61.3
8 1.0220365 4.0 65.3
9 0.9368778 3.7 69.0
10 0.8644932 3.4 72.4
11 0.7973206 3.1 75.5
12 0.7896729 3.1 78.6
13 0.7637169 3.0 81.6
14 0.6590202 2.6 84.2
15 0.6328695 2.5 86.7
16 0.6137555 2.4 89.1
17 0.4826804 1.9 91.0
18 0.4510860 1.8 92.8
19 0.3998378 1.6 94.4
20 0.3005401 1.2 95.6
21 0.2833715 1.1 96.7
22 0.2543039 1.0 97.7
23 0.2154730 0.8 98.5
24 0.1757374 0.7 99.2
25 0.1378358 0.5 99.7
26 0.0368678 0.1 99.8
prelim_pca$var_df %>% 
  ggplot(aes(x = Principle.component, y = Singular.values)) +
  geom_point() +
  geom_segment(aes(y = Singular.values, yend = 0, xend = Principle.component)) +
  theme_bw() +
  labs(x = "Principal component", y = "Singular value") +
  ggtitle("Singular values of preliminary PCA")

And next let’s examine the top few PCA loadings, corresponding to each of the first six principal components (PCs), or first six patterns governing our data:

prelim_pca$load

3.3 PCP parameter gridsearch

Now we’re about ready to run our first gridsearch.

D <- queens_standardized %>% 
  select(-Date) %>% 
  as.matrix() %>% 
  apply(., 2, mean_impute)

We can run a grid search using the default_eta and searching through all ranks 1 through 15 with the following code chunk:

pcp_defaults <- queens_standardized %>% select(-Date) %>% get_pcp_defaults()
default_eta <- pcp_defaults$eta

grid <- grid_search_cv(
  mat = D, 
  pcp_func = RRMC, 
  grid = data.frame(r = c(15)), 
  eta = default_eta
)

This takes quite a while, so we have saved the results to a .rds file using grid_search_cv’s save_as argument, which we load below:

grid <- readRDS(here::here("my-doc", "huge_eta_search.rds"))
grid$all_stats %>% 
  group_by(r, eta) %>% 
  summarize(
    avg_rel_err = mean(rel_err), 
    avg_sparsity = mean(S_sparsity), 
    avg_rank = mean(L_rank)
  ) %>% 
  arrange(avg_rel_err)
## # A tibble: 66 × 5
## # Groups:   r [6]
##        r   eta avg_rel_err avg_sparsity avg_rank
##    <int> <dbl>       <dbl>        <dbl>    <dbl>
##  1     3  0.2        0.846        100.         3
##  2     3  0.15       0.846         99.9        3
##  3     3  0.1        0.852         99.8        3
##  4     3  0.3        0.853        100.         3
##  5     3  0.25       0.856        100.         3
##  6     2  0.3        0.881        100.         2
##  7     3  0.4        0.882        100.         3
##  8     2  0.25       0.882        100.         2
##  9     3  0.35       0.882        100.         3
## 10     2  0.2        0.884        100.         2
## # ℹ 56 more rows
plot_mat <- function(D) {
  heatmaply::heatmaply(D, Rowv=F, Colv=F, showticklabels = FALSE, showtickmarks = FALSE)
}
rrmc_out <- RRMC(D, r = 3, eta = 0.1)
plot_mat(rrmc_out$L)
plot_mat(rrmc_out$S)
colnames(rrmc_out$L) <- colnames(D)
rrmc_pca <- pca(rrmc_out$L, pcs = paste("PC", 1:3, sep = ""))
rrmc_pca$var
Principle component Singular value % variance Total cumulative variance
1 5.206772 55.2 55.2
2 2.289758 24.3 79.5
3 1.942686 20.6 100.1
4 0.000000 0.0 100.1
5 0.000000 0.0 100.1
6 0.000000 0.0 100.1
7 0.000000 0.0 100.1
8 0.000000 0.0 100.1
9 0.000000 0.0 100.1
10 0.000000 0.0 100.1
11 0.000000 0.0 100.1
12 0.000000 0.0 100.1
13 0.000000 0.0 100.1
14 0.000000 0.0 100.1
15 0.000000 0.0 100.1
16 0.000000 0.0 100.1
17 0.000000 0.0 100.1
18 0.000000 0.0 100.1
19 0.000000 0.0 100.1
20 0.000000 0.0 100.1
21 0.000000 0.0 100.1
22 0.000000 0.0 100.1
23 0.000000 0.0 100.1
24 0.000000 0.0 100.1
25 0.000000 0.0 100.1
26 0.000000 0.0 100.1
rrmc_pca$load

print_patterns <- function(pats, colgroups = NULL, n = 1:6, pat_type = "pat", title = "") {
  
  if (!is.null(colgroups)) {
    colgroups <- colgroups %>% dplyr::rename(chem = !!names(colgroups)[1])
  } else {
    colgroups <- data.frame(chem = rownames(pats), group = "1")
  }
  
  grouping <- names(colgroups)[2]
  
  colnames(pats) <- paste0(pat_type, stringr::str_pad(1:ncol(pats), width = 2, pad = "0", side = "left"))
  
  pats.df <- pats %>% 
    tibble::as_tibble() %>% 
    dplyr::mutate(chem = colgroups[[1]]) %>%
    tidyr::pivot_longer(-chem, names_to = "pattern", values_to = "loading") %>%
    dplyr::right_join(., colgroups, by = "chem")

  pats.df$chem <- factor(as.character(pats.df$chem), levels = unique(as.character(pats.df$chem)))

  loadings <- pats.df %>%
    dplyr::filter(pattern %in% paste0(pat_type, stringr::str_pad(n, width = 2, pad = "0", side = "left"))) %>%
    ggplot(aes(x = chem, y = loading, color = !!sym(grouping))) +
    geom_point() +
    geom_segment(aes(yend=0, xend = chem)) +
    facet_wrap(~ pattern) + theme_bw() +
    theme(legend.position = "bottom",
          axis.text.x = element_text(angle = 45, hjust = 1),
          strip.background = element_rect(fill = "white"),
          axis.title.x = element_blank(),
          axis.title.y = element_blank()) +
    geom_hline(yintercept = 0, size = 0.2) + ggtitle(title)
  
  loadings
}
# for FA
library(psych)
library(GPArotation)
library(ggrepel)

rrmc_fa <- fa(rrmc_out$L, nfactors = 3, n.obs = nrow(rrmc_out$L), rotate = "varimax", scores = "regression")
## Warning in cor.smooth(r): Matrix was not positive definite, smoothing was done
print(rrmc_fa, digits = 2)
## Factor Analysis using method =  minres
## Call: fa(r = rrmc_out$L, nfactors = 3, n.obs = nrow(rrmc_out$L), rotate = "varimax", 
##     scores = "regression")
## Standardized loadings (pattern matrix) based upon correlation matrix
##       MR1   MR2   MR3 h2      u2 com
## Al  -0.05  0.98  0.20  1 0.00076 1.1
## NH4  1.00  0.09 -0.01  1 0.00036 1.0
## As   1.00  0.04  0.02  1 0.00038 1.0
## Ba  -0.18  0.97 -0.14  1 0.00086 1.1
## Br   0.95  0.16  0.27  1 0.00043 1.2
## Cd   0.83 -0.13 -0.55  1 0.00067 1.8
## Ca   0.77  0.60  0.23  1 0.00038 2.1
## Cl   0.10 -0.07  0.99  1 0.00137 1.0
## Cr   0.38  0.92  0.12  1 0.00056 1.4
## Cu   0.69  0.71 -0.16  1 0.00042 2.1
## EC   0.83  0.53 -0.17  1 0.00035 1.8
## Fe   0.56  0.82 -0.06  1 0.00047 1.8
## Pb   0.97  0.04 -0.26  1 0.00042 1.1
## Mg  -0.12  0.17  0.98  1 0.00128 1.1
## Mn   0.77  0.64  0.04  1 0.00035 1.9
## Ni   1.00  0.03  0.06  1 0.00039 1.0
## OC   0.77  0.63 -0.14  1 0.00037 2.0
## K    0.90  0.34  0.25  1 0.00038 1.5
## Se   0.97  0.00 -0.22  1 0.00042 1.1
## Si   0.06  0.98  0.16  1 0.00072 1.1
## Na   0.06  0.02  1.00  1 0.00135 1.0
## S    0.95  0.30  0.07  1 0.00032 1.2
## Ti  -0.06  1.00 -0.02  1 0.00079 1.0
## NO3  1.00 -0.03  0.05  1 0.00042 1.0
## V    0.99  0.10  0.10  1 0.00038 1.0
## Zn   1.00  0.02 -0.03  1 0.00039 1.0
## 
##                         MR1  MR2  MR3
## SS loadings           14.62 7.63 3.74
## Proportion Var         0.56 0.29 0.14
## Cumulative Var         0.56 0.86 1.00
## Proportion Explained   0.56 0.29 0.14
## Cumulative Proportion  0.56 0.86 1.00
## 
## Mean item complexity =  1.3
## Test of the hypothesis that 3 factors are sufficient.
## 
## df null model =  325  with the objective function =  523.71 with Chi Square =  798399
## df of  the model are 250  and the objective function was  331.99 
## 
## The root mean square of the residuals (RMSR) is  0 
## The df corrected root mean square of the residuals is  0 
## 
## The harmonic n.obs is  1535 with the empirical chi square  0.1  with prob <  1 
## The total n.obs was  1535  with Likelihood Chi Square =  505460.2  with prob <  0 
## 
## Tucker Lewis Index of factoring reliability =  0.176
## RMSEA index =  1.147  and the 90 % confidence intervals are  1.145 1.15
## BIC =  503626.1
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                   MR1 MR2 MR3
## Correlation of (regression) scores with factors     1   1   1
## Multiple R square of scores with factors            1   1   1
## Minimum correlation of possible factor scores       1   1   1
loadings <- as.data.frame(cbind(rownames(rrmc_fa$loadings[]), rrmc_fa$loadings[])) %>% rename(Variable = V1) 

loadings <- loadings %>% mutate_at(colnames(loadings)[str_starts(colnames(loadings), "MR")], as.numeric)

loadings$Max <- colnames(loadings[, -1])[max.col(loadings[, -1], ties.method = "first")] # should be 2:5

loadings %>% kableExtra::kbl(caption = "Loadings") %>% kableExtra::kable_classic(full_width = F, html_font = "Cambria", position = "center") %>% 
  kableExtra::kable_styling(bootstrap_options = c("hover", "condensed"), fixed_thead = T) %>% kableExtra::scroll_box(width = "100%", height = "400px")
Table 3.1: Loadings
Variable MR1 MR2 MR3 Max
Al Al -0.0516901 0.9774711 0.2027738 MR2
NH4 NH4 0.9960585 0.0862511 -0.0080859 MR1
As As 0.9986695 0.0438255 0.0189229 MR1
Ba Ba -0.1774868 0.9735461 -0.1408622 MR2
Br Br 0.9507836 0.1590576 0.2651090 MR1
Cd Cd 0.8255029 -0.1342269 -0.5475927 MR1
Ca Ca 0.7684500 0.5971629 0.2291292 MR1
Cl Cl 0.0989384 -0.0731368 0.9917119 MR3
Cr Cr 0.3775964 0.9180500 0.1185138 MR2
Cu Cu 0.6862436 0.7090304 -0.1610070 MR2
EC EC 0.8327887 0.5261587 -0.1710746 MR1
Fe Fe 0.5617517 0.8248882 -0.0593728 MR2
Pb Pb 0.9652234 0.0399142 -0.2575474 MR1
Mg Mg -0.1198352 0.1699795 0.9774787 MR3
Mn Mn 0.7711625 0.6351324 0.0396162 MR1
Ni Ni 0.9978816 0.0260774 0.0562011 MR1
OC OC 0.7672172 0.6266471 -0.1353504 MR1
K K 0.9043054 0.3418836 0.2548839 MR1
Se Se 0.9745303 -0.0042609 -0.2232732 MR1
Si Si 0.0617839 0.9846615 0.1609551 MR2
Na Na 0.0639449 0.0171711 0.9971282 MR3
S S 0.9521291 0.2981006 0.0653146 MR1
Ti Ti -0.0596583 0.9976637 -0.0178691 MR2
NO3 NO3 0.9981291 -0.0343629 0.0462689 MR1
V V 0.9901870 0.0956123 0.1000473 MR1
Zn Zn 0.9992008 0.0157969 -0.0310069 MR1
scores <- as.data.frame(cbind(rownames(rrmc_fa$scores[]), rrmc_fa$scores[])) %>% mutate_all(as.numeric)

scores$Max <- colnames(scores)[max.col(scores, ties.method = "first")]

scores %>% 
  kableExtra::kbl(caption = "Scores") %>% 
  kableExtra::kable_classic(full_width = F, html_font = "Cambria", position = "center") %>% 
  kableExtra::kable_styling(bootstrap_options = c("hover", "condensed"), fixed_thead = T) %>% 
  kableExtra::scroll_box(width = "100%", height = "400px")
Table 3.1: Scores
MR1 MR2 MR3 Max
0.0810764 -0.2148536 4.7956037 MR3
-0.1758763 -0.8842593 -0.4744605 MR1
-0.5228369 -0.5805808 -0.2534992 MR3
3.7969090 -0.1811318 0.1175294 MR1
0.0170107 -0.2034000 -0.3337588 MR1
0.3250447 -0.2863093 0.5386281 MR3
-0.4663021 0.0959024 0.2860500 MR3
2.0423722 -0.7021320 -0.2626516 MR1
2.2286770 0.2290778 0.5767719 MR1
0.2633727 0.6887223 -0.2459222 MR2
1.8155311 0.2395212 0.7728694 MR1
1.0769308 -0.8702415 -0.1874851 MR1
0.5266173 0.3533616 0.5098058 MR1
-0.8160942 -0.5398569 -0.4457002 MR3
1.2113353 0.0682394 -0.4312160 MR1
0.7640474 -1.2935247 0.1144776 MR1
0.4466886 1.0627672 -0.1442057 MR2
1.0779673 1.3788768 0.2897110 MR2
0.4181035 0.9335465 -0.4760338 MR2
0.3750419 -0.1395150 -0.5524119 MR1
0.2514821 0.3204234 -0.4949604 MR2
0.1617738 0.0434261 -0.5974609 MR1
0.0420542 -0.1488882 -0.6347192 MR1
1.2263231 -0.5701449 -0.2421441 MR1
-0.3861805 -0.6038678 -0.3965881 MR1
-0.1971249 -0.7570192 1.8056564 MR3
-0.7487152 -0.8230022 -0.4429393 MR3
0.1280251 0.1133143 -0.6027645 MR1
0.3094131 -0.0662733 -0.5007059 MR1
2.0766335 0.3415935 -0.3133643 MR1
0.4801154 0.3321426 -0.4596223 MR1
1.1938788 -0.2405081 0.0319784 MR1
-0.3752192 -0.0473881 -0.4340776 MR2
-0.3469356 -0.5764211 -0.4363891 MR1
0.1316144 0.1480555 -0.3616684 MR2
0.7538958 -0.4999452 -0.3166927 MR1
0.0738393 -0.6917066 -0.4745045 MR1
-0.2114867 -0.2322134 -0.5205919 MR1
0.9764958 0.3764385 0.2244427 MR1
0.3707445 0.2158998 -0.5639781 MR1
-0.0026864 -0.5582212 -0.1673292 MR1
1.7355883 0.2183385 -0.5025919 MR1
-0.6954272 -0.4709416 -0.5881040 MR2
0.5312585 1.2349601 -0.5213315 MR2
0.9811824 0.4687826 -0.6511636 MR1
1.7426769 0.1378031 0.4326821 MR1
1.6559280 1.1507623 -0.0450253 MR1
-0.0541048 1.4125299 -0.1146895 MR2
1.0567550 0.4067217 -0.5909504 MR1
-1.6517787 3.3544503 -0.1153964 MR2
-0.7913998 -0.2904158 -0.6354265 MR2
-0.0347403 0.1577791 -0.6542762 MR2
-0.3987341 -0.7523689 0.0352195 MR3
-0.4186711 -0.0036712 0.0690681 MR3
-0.6140767 -0.5144627 -0.5385686 MR2
-0.4940698 -0.4694065 0.0067375 MR3
-0.8194453 -0.2367139 -0.4431056 MR2
0.1140868 -0.1304596 -0.0188363 MR1
-0.8874513 -0.2687159 -0.5693483 MR2
0.4453676 -0.0974445 0.0832252 MR1
-0.2657420 -0.4785598 -0.2827901 MR1
1.1043230 -0.0967694 -0.3950461 MR1
-0.4335108 -0.4511239 -0.1482413 MR3
0.1389915 -0.3255591 -0.6686203 MR1
0.9104198 0.2452368 -0.5984947 MR1
-0.6732931 -0.8213583 -0.4908518 MR3
1.7433415 0.7342026 0.0122654 MR1
3.7598928 1.1595188 -0.4394638 MR1
-0.5639907 -0.7303658 -0.5329985 MR3
-0.7531444 -0.8237931 -0.5765816 MR3
-0.2864436 -1.1418485 4.6343674 MR3
0.5970887 0.4595038 0.0601781 MR1
0.2319998 -0.6629018 -0.2543002 MR1
6.1124923 1.2484408 -0.1782272 MR1
-0.8604220 -0.8825582 0.3283628 MR3
0.5684062 1.2106437 -0.8168400 MR2
-0.1544079 -0.0526669 -0.1493026 MR2
0.9780299 -0.2324845 -0.3881576 MR1
-0.4862931 -0.9292692 -0.3154338 MR3
-0.0614555 -0.9572409 -0.5287899 MR1
0.3419955 1.6183260 -0.5078970 MR2
-0.5716393 -0.9186781 0.8424933 MR3
1.4938053 -1.0921639 -0.1911225 MR1
0.7880810 -0.4325646 0.0990783 MR1
1.7980323 -0.0937914 -0.0899182 MR1
2.2078315 -0.8184787 -0.3522727 MR1
0.7306635 -0.6848714 -0.3526078 MR1
1.2878815 -0.6606952 0.0515308 MR1
0.7837497 0.0404739 -0.3635784 MR1
0.2994667 -0.1671932 0.2968346 MR1
2.0867267 -0.6600112 0.0539197 MR1
0.8232699 -0.6603968 0.1998122 MR1
1.2219683 -0.7524162 0.3354554 MR1
2.2995283 -0.6472942 0.0994590 MR1
2.4249858 -0.1537253 0.3788635 MR1
5.6500716 0.2592440 0.4588566 MR1
2.5105456 0.2289599 -0.1965432 MR1
2.5272708 -0.1097335 -0.4613124 MR1
1.0095978 -0.2681602 0.1382686 MR1
0.2405660 -0.6338439 -0.3383098 MR1
-0.0925203 -0.7759681 -0.0541757 MR3
2.1611230 -0.8010057 -0.0149140 MR1
0.6237412 -0.3196673 -0.4936859 MR1
0.3157937 -0.7538374 -0.0189634 MR1
0.4520401 -0.5387934 0.0127517 MR1
0.6225843 -0.5806427 -0.1626974 MR1
1.9551020 -1.1626441 -0.3105256 MR1
-0.0385554 -0.7986454 -0.3229689 MR1
0.1314752 -0.1334263 -0.3948273 MR1
0.2460156 0.1456481 -0.1363526 MR1
4.2679893 -1.0302204 0.6146390 MR1
-0.5295391 -0.6005388 -0.1809599 MR3
0.0088405 0.0199748 -0.3102921 MR2
1.2322148 0.0297614 -0.0834964 MR1
-0.1176026 -0.8931884 1.1135294 MR3
-0.7400349 -1.0051236 -0.5618719 MR3
3.1083903 2.3694491 0.0944544 MR1
1.8413000 0.3834886 0.9146489 MR1
-0.8362622 0.6221428 -0.3860755 MR2
-0.3620831 -0.6678118 -0.1031946 MR3
-0.0782141 -0.1126455 0.0760418 MR3
1.1430711 -0.5149046 -0.1972346 MR1
1.5512042 1.7262285 -0.0396571 MR2
1.9722680 1.8428490 3.4201319 MR3
0.1824578 1.4932779 -0.1959038 MR2
1.1819318 1.5910090 -0.2929092 MR2
-0.0940834 -0.9589320 -0.4221880 MR1
-0.0045319 0.9022000 -0.4981604 MR2
-0.6874488 2.6864109 0.0494737 MR2
-0.6594410 -0.3754517 -0.4839320 MR2
-0.4304907 0.9077372 -0.2220744 MR2
2.0790435 1.7950628 0.5559775 MR1
0.7157535 0.8010541 -0.1879820 MR2
-0.0389399 -0.0463574 0.0465240 MR3
-0.0135926 0.1161491 -0.3433647 MR2
1.6764088 -0.3513837 -0.2661990 MR1
0.1706095 1.7126882 -0.9286231 MR2
0.1007205 0.8664840 -0.7046681 MR2
0.1629021 -0.6613571 0.4145853 MR3
0.8336824 1.9344255 0.2051441 MR2
-0.1785886 -0.5792182 0.8885868 MR3
1.4221290 -0.0031892 0.0522870 MR1
0.8570131 0.8547118 -0.5042495 MR1
-0.4263566 -0.3332302 -0.5632275 MR2
1.0986987 -0.3681083 -0.6706239 MR1
0.5873167 -0.2425405 -0.1960892 MR1
0.1236177 -0.0664483 0.1611528 MR3
0.2399607 -0.2793851 -0.4348648 MR1
0.2964210 -0.0747946 -0.3980851 MR1
-0.2689544 -0.0567702 -0.5438300 MR2
1.0531220 1.0286225 -0.1111870 MR1
-0.7045690 -0.3208099 -0.4937162 MR2
0.9081174 1.7188767 0.5267676 MR2
1.0099941 1.2839801 0.0013430 MR2
-0.7786522 -0.0285106 -0.1805548 MR2
0.1145196 0.0346599 0.6357661 MR3
1.6461233 0.1483980 0.2888282 MR1
0.4845109 0.9504312 -0.3953054 MR2
-0.6414240 0.0151119 -0.6057159 MR2
-0.3934751 1.2502947 -0.2557944 MR2
0.5104198 0.1212605 0.1602646 MR1
-0.5897725 -0.4526990 -0.5754374 MR2
0.1869337 0.1886494 -0.1074033 MR2
1.3712303 0.7813277 -0.6034153 MR1
-0.5159564 -0.0352466 0.0655931 MR3
-0.3814615 -0.9908138 -0.1205334 MR3
0.3627768 -0.0284462 -0.6983198 MR1
-0.4125468 -0.6916682 -0.5035996 MR1
-0.9423026 -0.9379775 -0.3117143 MR3
-0.4885021 -0.2416682 -0.7451364 MR2
1.3231365 1.5250067 -0.7655793 MR2
2.4304159 0.8182932 0.2399433 MR1
-0.4418449 -0.6739767 -0.4248096 MR3
-0.2220484 0.7661201 -0.3150759 MR2
-0.7965088 -0.2310766 -0.4055035 MR2
1.4152521 0.0747046 -0.7900512 MR1
-0.0395171 -0.4326002 -0.4131414 MR1
-0.5392898 -0.6693919 -0.5630151 MR1
0.1878218 -0.1756450 -0.1728451 MR1
-0.6656194 -0.9971919 -0.3345037 MR3
0.7127787 0.2322898 -0.3580389 MR1
-0.6879988 -0.6146689 -0.6229453 MR2
-0.6028200 -0.7276010 -0.5728823 MR3
-0.7040649 1.6837126 -0.4578802 MR2
1.2543299 1.2953016 -0.4776841 MR2
0.6966846 1.2413494 -0.5263223 MR2
1.2950087 -0.3064430 4.3561508 MR3
2.0583171 0.6019274 -0.5031652 MR1
1.4254560 0.1922260 -0.3868243 MR1
0.1631737 -0.5377688 0.0011925 MR1
-0.5628498 -0.7565792 -0.2501899 MR3
-0.1231969 -0.4363394 -0.2692232 MR1
-0.0450966 -0.5891930 -0.3805321 MR1
-0.1794390 -0.8028956 -0.3431714 MR1
0.1092545 -0.1054115 -0.3363054 MR1
1.0870164 2.6530662 -0.3362178 MR2
1.2032864 -1.0789124 0.9596732 MR1
0.6228146 -0.2529101 -0.1468190 MR1
1.3291072 -0.2134362 0.6583801 MR1
0.3845415 -0.2451617 -0.3222264 MR1
0.2059188 -0.6486274 -0.4347729 MR1
-0.4703339 -1.1278500 3.0577949 MR3
0.3396781 -0.9126818 -0.2131008 MR1
-0.2179310 -0.4411394 -0.2701635 MR1
1.2635961 -0.0518947 0.1141387 MR1
0.4979108 -0.8409490 -0.2312993 MR1
2.5388070 -0.3565396 -0.1037327 MR1
3.7942105 -0.7177197 -0.3821870 MR1
0.5860241 -0.3889473 -0.5785011 MR1
1.0450394 -0.2936426 0.1222715 MR1
2.6105817 -0.8171003 0.3236371 MR1
8.8558543 -0.2418781 1.8745648 MR1
4.5123853 -0.4452649 0.8411076 MR1
3.1246607 -0.6345444 0.3664875 MR1
0.1319661 -0.7112776 -0.2268638 MR1
1.3992681 -0.7919444 0.2430627 MR1
2.7546988 -0.4253006 1.2059527 MR1
3.3019241 -0.1393201 -0.4694823 MR1
1.5875123 -0.7800069 -0.4515860 MR1
3.0273585 -0.3704171 -0.0180569 MR1
0.1682228 0.1324543 -0.0906750 MR1
1.5136990 0.1558751 -0.2116527 MR1
0.4264055 -0.0429628 -0.0200022 MR1
2.6171032 -1.1616082 0.1512460 MR1
0.7744855 -0.8086920 -0.2361216 MR1
3.2406949 -0.0678503 0.4788734 MR1
0.2678950 0.1832925 0.0299888 MR1
4.3654850 1.3684571 0.3142868 MR1
0.4107358 -0.5072909 1.2979051 MR3
1.8644404 0.6618966 0.7352134 MR1
0.7852158 -0.5567173 -0.2659279 MR1
0.1704379 0.0748278 -0.3201578 MR1
0.0744399 -0.2754846 0.4026193 MR3
-0.2050961 -0.3373218 -0.1715237 MR3
-0.2968660 -1.3003378 8.7136521 MR3
0.3844010 -0.7687465 -0.0101821 MR1
0.2111862 -0.2692736 -0.4270621 MR1
-0.4503988 -0.6555110 -0.4449079 MR3
0.2598094 0.2715178 -0.3804887 MR2
0.3168486 -0.4956262 -0.2044324 MR1
-0.1937094 -0.4380638 -0.3182840 MR1
-0.6849791 -0.5774941 -0.4817179 MR3
0.2051122 -0.2665339 -0.2167265 MR1
0.7611227 -0.1470331 -0.1448832 MR1
-0.5894375 0.2622500 -0.5535449 MR2
-0.4308771 -0.5824177 0.7340618 MR3
0.1924050 0.7668244 0.0459657 MR2
-0.4465059 -0.3307522 -0.5374860 MR2
1.8832200 -0.2010530 0.2436684 MR1
0.1540543 -0.8605192 0.7404922 MR3
-0.2975433 -0.7073806 -0.4670457 MR1
0.8796060 -0.7108633 0.8214977 MR1
1.0669886 0.0406871 0.4411244 MR1
-0.5371708 -0.3260840 -0.4812387 MR2
-0.5463071 0.1868416 0.2646801 MR3
0.4530652 -0.4428407 1.5779595 MR3
-0.7583031 -0.6994451 0.6966485 MR3
2.5911665 -0.2758498 -0.5790147 MR1
1.6517177 0.2662951 0.1043178 MR1
2.1542517 0.1020534 1.4752378 MR1
0.4003159 0.7852613 -0.0949435 MR2
-0.8029629 0.4358520 -0.4223164 MR2
-0.9014336 -0.7406472 -0.1210750 MR3
2.5643009 1.9701221 -0.1657794 MR1
-0.6698272 -0.3727789 -0.2103756 MR3
-0.6412432 -0.5945272 -0.2778974 MR3
0.2545756 -0.1921685 0.2403679 MR1
0.1039545 0.0054017 -0.6418658 MR1
1.3321965 0.0298186 -0.4843414 MR1
0.1327031 -0.5323905 -0.5273875 MR1
-0.1432602 0.4607961 -0.5181339 MR2
0.1462176 0.2901067 0.1923919 MR2
0.1791141 0.8381455 -0.4281444 MR2
0.2723080 0.2843142 -0.1663244 MR2
0.5973961 0.0189579 -0.0954542 MR1
1.3795615 0.2042003 -0.4140796 MR1
-0.8654603 -0.2404233 -0.4456684 MR2
0.2587406 -0.7457814 0.2366388 MR1
1.8810312 0.3784359 -0.3154466 MR1
0.8295934 2.0214966 -0.2378713 MR2
0.5933806 -0.1547266 -0.2285107 MR1
0.9514459 -0.0965601 -0.2488947 MR1
0.5177101 0.7452668 -0.5030058 MR2
-0.2349736 -0.3471821 -0.1030284 MR3
-1.0838415 -0.9058636 -0.6006952 MR3
0.4153132 0.0631519 -0.5668814 MR1
0.4513308 2.1300343 -0.3487474 MR2
-0.4869582 -0.2293756 -0.5783945 MR2
0.4264398 -0.0292687 -0.2443509 MR1
-0.5064732 -0.3033510 -0.5639460 MR2
-0.1746232 0.5307425 0.5888776 MR3
-0.8902565 -1.0890876 2.2615328 MR3
0.8510291 1.1433297 -0.2508911 MR2
0.2610042 0.1287409 0.2006784 MR1
-0.2233585 0.3595791 -0.6315433 MR2
1.8797745 -0.1425529 0.2347231 MR1
0.9585337 -0.0606244 -0.3675955 MR1
-0.8025637 -0.5758678 -0.3142294 MR3
1.3160687 -0.3790952 0.0474074 MR1
0.3066448 -0.4399113 -0.2932203 MR1
0.1893144 -0.2158766 -0.3885193 MR1
0.9881218 0.2695058 0.3725051 MR1
-0.3586681 -0.6773104 -0.3855770 MR1
0.4910487 1.0676056 -0.7568542 MR2
2.0310011 2.7823966 -0.5002178 MR2
-0.7350271 -0.5610106 0.3398527 MR3
0.2208306 -0.7282578 -0.5468778 MR1
1.4725023 0.1855094 -0.3269514 MR1
-0.2492757 0.3071318 -0.2072599 MR2
2.1195293 -0.5662549 -0.2666153 MR1
0.4280426 -0.3073045 -0.4195995 MR1
0.3764864 -0.2814926 -0.3409435 MR1
2.7905255 0.2949316 -0.2441495 MR1
0.4806014 0.2291202 -0.3088816 MR1
4.3216584 2.0017425 0.3765289 MR1
-0.3341741 -0.7126095 -0.4711623 MR1
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-0.5436459 1.5700404 -0.8357402 MR2
-1.0469104 -0.3222221 0.0715230 MR3
-0.1378527 1.4796639 -0.4417236 MR2
-0.7265324 0.2440331 -0.8507889 MR2
-0.1381405 2.3639121 -0.1776378 MR2
-0.7221379 0.3015281 -0.3558042 MR2
-1.0195266 0.5552086 -0.4259692 MR2
-0.7945591 0.4599741 -0.4984870 MR2
-0.5166759 -0.5485588 -0.3148669 MR3
0.2224848 1.0586927 -0.2829445 MR2
-0.6601676 0.8383087 0.6766019 MR2
-1.0070172 0.5229743 -0.0404542 MR2
-1.1502283 0.8196942 -0.1828275 MR2
-0.3316134 0.1354834 -0.0072292 MR2
0.3491894 2.3810778 -0.2840717 MR2
-0.7943375 -0.9834609 1.2523271 MR3
-0.1853149 0.1986097 -0.0588846 MR2
-1.1487299 0.4740112 -0.4232972 MR2
-1.2256098 2.1500543 -0.1168082 MR2
-0.8365817 0.4647807 -0.6504027 MR2
-0.9153940 2.3079543 0.0905829 MR2
-1.1734118 -0.0496448 0.4092620 MR3
-1.2528497 0.5867340 -0.7013062 MR2
-1.1146435 -0.2568470 2.8068390 MR3
-0.5620033 0.5940168 -1.0543719 MR2
-0.7692963 2.7670330 0.4946943 MR2
-1.1852102 2.0744447 0.2338451 MR2
-0.7193087 0.5267360 1.0571569 MR3
-0.2681011 1.7708039 -0.9210606 MR2
-1.2612487 -0.8227920 3.5516037 MR3
-0.6825450 0.4192213 0.0024144 MR2
-0.6540195 0.0779745 0.7048850 MR3
-1.0799796 0.8747948 -0.6351929 MR2
-1.1892840 2.9158654 -0.4122505 MR2
-0.9432902 0.9507674 2.1355484 MR3
-0.7783719 -0.3451348 -0.4765812 MR2
-0.9224370 -0.1014693 0.3728341 MR3
-0.3121997 0.0849950 -0.0738019 MR2
-0.2985321 1.5950594 -0.4638251 MR2
1.3806769 6.8602496 -1.1114227 MR2
-0.7022650 0.0999142 1.9090310 MR3
-1.0393944 1.3961894 -0.6062689 MR2
-0.1136260 1.2081710 0.2909533 MR2
-0.6802390 -0.4543148 -0.1593873 MR3
-0.6941744 0.4157125 -0.1879515 MR2
-1.0987098 -0.4407950 0.1144636 MR3
-0.0832778 0.9887124 -0.6734708 MR2
-0.5780380 -0.4878207 -0.5810847 MR2
-0.2781742 -0.3203201 1.3400103 MR3
0.5388933 -0.4294215 -0.5782436 MR1
-1.2606089 -0.2692241 -0.6257414 MR2
-0.9369646 -0.1244716 1.3305741 MR3
0.0426521 -0.0949825 -0.5829014 MR1
1.6546003 1.5852446 0.1206034 MR1
-0.0230435 -0.4060361 -0.8721739 MR1
-0.7659360 -0.5693170 0.0978094 MR3
1.4006359 -1.1409185 -0.7846840 MR1
fa_pats <- loadings %>% 
  select(-Max, -Variable) %>% 
  mutate_all(as.numeric)

fa_pats <- fa_pats %>% select(sort(colnames(.))) %>% as.matrix()
print_patterns(fa_pats, pat_type = "factor", n = 1:3, title = "FA factors")
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

scores %>% dplyr::select(-Max) %>% corr.test() %>% print(short=FALSE)
## Call:corr.test(x = .)
## Correlation matrix 
##     MR1 MR2 MR3
## MR1   1   0   0
## MR2   0   1   0
## MR3   0   0   1
## Sample Size 
## [1] 1535
## Probability values (Entries above the diagonal are adjusted for multiple tests.) 
##      MR1 MR2 MR3
## MR1 0.00   1   1
## MR2 0.99   0   1
## MR3 1.00   1   0
## 
##  Confidence intervals based upon normal theory.  To get bootstrapped values, try cor.ci
##         raw.lower raw.r raw.upper raw.p lower.adj upper.adj
## MR1-MR2     -0.05     0      0.05  0.99     -0.06      0.06
## MR1-MR3     -0.05     0      0.05  1.00     -0.05      0.05
## MR2-MR3     -0.05     0      0.05  1.00     -0.06      0.06
loadings %>% 
  ggplot(aes(x = MR1, y = MR2)) + 
  geom_point() + geom_label_repel(aes(label = Variable),
                                  box.padding   = 0.35,
                                  point.padding = 0.5,
                                  segment.color = 'grey50') + 
  theme(legend.position = "bottom") +
  labs(title = "Variable Loadings on First and Second Factors")

if ("MR3" %in% colnames(loadings)) {
  loadings %>% 
  ggplot(aes(x = MR1, y = MR3)) + 
  geom_point() + geom_label_repel(aes(label = Variable),
                                  box.padding   = 0.35,
                                  point.padding = 0.5,
                                  segment.color = 'grey50') + 
  theme(legend.position = "none") +
  labs(title = "Variable Loadings on First and Third Factors")
}

plot_loadings <- loadings %>% select(-Max) %>% gather(key = "Factor", value = "Loading", -Variable) %>% mutate(Factor = str_replace(Factor, "MR", "Factor "))

plot_loadings %>% 
  ggplot(aes(x = Factor, y = Loading, fill = Factor)) + geom_col(position = "dodge") +
  facet_wrap(~ Variable) + 
  theme(legend.position = "none", axis.text.x = element_text(angle = 45, hjust = 1)) +
  coord_flip() + geom_hline(yintercept = 0, size = 0.2) +
  labs(title = "Variable Loadings on All Factors")

scores %>% ggplot(aes(x = Max, fill = Max)) + geom_bar() +
  labs(x = "Factors", y = "Number of Individuals", title = "Number with Highest Scores per Factor") +
  theme(legend.position = "none")

scores %>% group_by(Max) %>% summarise(n())
## # A tibble: 3 × 2
##   Max   `n()`
##   <chr> <int>
## 1 MR1     522
## 2 MR2     531
## 3 MR3     482
scores %>% gather(key = "factor", value = "score", -Max) %>% dplyr::select(-Max) %>% 
  ggplot(aes(x = score)) + geom_density() + facet_grid(factor~.) # should be MR1:MR4

library(NMF)

num_cores <- ceiling(parallel::detectCores() / 2)
nmf_mat <- rrmc_out$L
nmf_mat[nmf_mat < 0] <- 0
res <- nmf(rrmc_out$L, rank = 3, method = "offset", nrun = 30, seed = 0, .opt = paste0('vp', num_cores))

# only compute the scores
s <- featureScore(res)
summary(s)
# compute the scores and characterize each metagene
s <- extractFeatures(res) 
str(s)

W <- basis(res) # basis matrix / metagenes / contribution matrix
H <- coef(res) # mixture coeffecient matrix / metagene expression profiles / profile matrix ie. loadings ?

heatmaply(W, main = "NMF basis / contribution matrix (scores)", Rowv = F, Colv = F, 
          ylab = params$rowvar_name, labRow = as.character(rowlabs),
          cexRow = 100, row_side_colors = data.frame("cohort" = cohorts), showticklabels = c(T, F))

heatmaply(H, main = "NMF profile matrix (loadings)", Rowv = F, Colv = F,
          col_side_colors = data.frame("exposure family" = as.factor(params$colgroupings)))

loadings <- t(H) %>% as.tibble()

print_patterns(loadings, colgroups = cng, title = "NMF loadings")